The finite dimensional analog of this statement, set in \(\mathbb {R}^n\)  with the prefactor \((1+|x|)^{-1}\) , is presented in [1]}. Moreover, for \(\mathbb {R}^\mathbb {N}\) , in [2]} the condition reads as
\(\int _0^T \int |p_i(c_t)|\,\mathrm {d}\nu _t \, \mathrm {d}t < \infty , \qquad \mbox{for all $i\in \mathbb {N}$}.\) 

